question is
|x|+|x-1|=1 or |x-1|=1-|x|

this is possible only when x is positive or zero.
From 1 x>=0..sufficient
From 2 x<=1..but it is not clear that whether x>0...not sufficient

hence ans option A

C

There will be 3 ranges:

1) x<0
-x-x+1=1

2) 0 <=x < 1
x-x-1=1

3)x>1
x+x-1=2x-1

St. 1 -- insuff
St. 2 -- insuff

St. 1 & St. 2 together become 0 <= x <=1 which is sufficient

What's the source? I think the OA is wrong.

Statement 1: If we use 5 as the value of x since it conforms to the constraints of being greater than or equal to 0, then |5| + |5 -1| =/ 1

Therefore, statement one is insufficient.

Statement 2: Same thing just plug in negative 5. Therefore, insufficient.

When you combine the two statements you learn that 0<= x <= 1

You can test all values for this range and the statement |x| + |x -1| = 1 is true. I would choose C

hmm..all this time I was thinking only with fractions..missed the scenario for x>1 all togather.Thanks to osirus0830 and jube for pointing out.
A definitely cannot be the ans..same goes for B

I agree with C as for any value of 0<=x<1..|x|+|x-1|=1 will be true